# zbMATH — the first resource for mathematics

Model approximation for two-dimensional Markovian jump systems with state-delays and imperfect mode information. (English) Zbl 1349.62381
Summary: This paper is concerned with the problem of $$\mathcal {H}_{\infty}$$ model approximation for a class of two-dimensional (2-D) discrete-time Markovian jump linear systems with state-delays and imperfect mode information. The 2-D system is described by the well-known Fornasini-Marchesini local state-space model, and the imperfect mode information in the Markov chain simultaneously involves the exactly known, partially unknown and uncertain transition probabilities. By using the characteristics of the transition probability matrices, together with the convexification of uncertain domains, a new $$\mathcal {H}_{\infty}$$ performance analysis criterion for the underlying system is firstly derived, and then two approaches, namely, the convex linearisation approach and iterative approach, to the $$\mathcal {H}_{\infty}$$ model approximation synthesis are developed. The solutions to the problem are formulated in terms of strict linear matrix inequalities (LMIs) or a sequential minimization problem subject to LMI constraints. Finally, simulation studies are provided to illustrate the effectiveness of the proposed design methods.

##### MSC:
 62M05 Markov processes: estimation; hidden Markov models
Full Text:
##### References:
 [1] Boukas, E. K. (2005). Stochastic switching systems: Analysis and design. Boston: Birkhauser. [2] Souza, CE, Robust stability and stabilization of uncertain discrete-time Markovian jump linear systems, IEEE Transactions on Automatic Control, 51, 836-841, (2006) · Zbl 1366.93479 [3] Ghaoui, L; Oustry, F; AitRami, M, A cone complementarity linearization algorithm for static output-feedback and related problems, IEEE Transactions on Automatic Control, 42, 1171-1176, (1997) · Zbl 0887.93017 [4] Gao, H; Lam, J; Wang, C, Model simplification for switched hybrid systems, Systems & Control Letters, 55, 1015-1021, (2006) · Zbl 1120.93311 [5] Gao, H; Lam, J; Wang, C; Wang, Q, Hankel norm approximation of linear systems with time-varying delay: continuous and discrete cases, International Journal of Control, 77, 1503-1520, (2004) · Zbl 1067.93008 [6] Ghafoor, A; Sreeram, V, Model reduction via limited frequency interval gramians, IEEE Transactions on Circuits and Systems I: Regular Papers, 55, 2806-2812, (2008) [7] Hoang, NT; Tuan, HD; Nguyen, TQ; Hosoe, S, Robust mixed generalized $${\mathscr {H}}_{2}/{\mathscr {H}}_{∞ }$$ filtering of 2-D nonlinear fractional transformation systems, IEEE Transactions on Signal Processing, 53, 4697-4706, (2005) · Zbl 1370.93277 [8] Kaczorek, T. (1985). Two-dimensional linear systems. Berlin: Springer. · Zbl 0593.93031 [9] Karan, M; Shi, P; Kaya, CY, Transition probability bounds for the stochastic stability robustness of continuous- and discrete-time Markovian jump linear systems, Automatica, 42, 2159-2168, (2006) · Zbl 1104.93056 [10] Lam, J; Gao, H; Xu, S; Wang, C, $$\mathscr {H}_{∞ }$$ and $$\mathscr {L}_{2}$$-$$\mathscr {L}_{∞ }$$ model reduction for system input with sector nonlinearities, Journal of Optimization Theory and Applications, 125, 137-155, (2005) · Zbl 1062.93020 [11] Lam, J; Xu, S; Zou, Y; Lin, Z; Galkowski, K, Robust output feedback stabilization for two-dimensional continuous systems in Roesser form, Applied Mathematics Letters, 17, 1331-1341, (2004) · Zbl 1122.93405 [12] Lin, Z, Feedback stabilization of MIMO 3-D linear systems, IEEE Transactions on Automatic Control, 44, 1950-1955, (1999) · Zbl 0956.93055 [13] Lin, Z; Lam, J; Galkowski, K; Xu, S, A constructive approach to stabilizability and stabilization of a class of nd systems, Multidimensional Systems and Signal Processing, 12, 329-343, (2001) · Zbl 1009.93042 [14] Liu, H; Ho, DWC; Sun, F, Design of $$\mathscr {H}_{∞ }$$ filter for Markov jumping linear systems with non-accessible mode information, Automatica, 44, 2655-2660, (2008) · Zbl 1155.93432 [15] Lu, W. S., & Antoniou, A. (1992). Two-dimensional digital filters. New York: Marcel Dekker. · Zbl 0852.93001 [16] Paszke, W; Lam, J; Galkowski, K; Xu, S; Lin, Z, Robust stability and stabilisation of 2D discrete state-delayed systems, Systems & Control Letters, 51, 277-291, (2004) · Zbl 1157.93472 [17] Peng, D; Guan, X, $${\mathscr {H}}_{∞ }$$ filtering of 2-D discrete state delayed systems, Multidimensional Systems and Signal Processing, 20, 265-284, (2009) · Zbl 1169.93363 [18] Qiu, J; Feng, G; Gao, H, Observer-based piecewise affine output feedback controller synthesis of continuous-time T-S fuzzy affine dynamic systems using quantized measurements, IEEE Transactions on Fuzzy Systems, 20, 1046-1062, (2012) [19] Roesser, RP, A discrete state-space model for linear image processing, IEEE Transactions on Automatic Control, 20, 1-10, (1975) · Zbl 0304.68099 [20] Wang, G; Zhang, Q; Sreeram, V, Design of reduced-order $$\mathscr {H}_{∞ }$$ filtering for Markovian jump systems with mode-dependent time delays, Signal Processing, 89, 187-196, (2009) · Zbl 1155.94337 [21] Wang, G; Zhang, Q; Sreeram, V, Partially mode-dependent $$\mathscr {H}_{∞ }$$ filtering for discrete-time Markovian jump systems with partly unknown transition probabilities, Signal Processing, 90, 548-556, (2010) · Zbl 1177.93086 [22] Wu, L; Shi, P; Gao, H; Wang, C, $$\mathscr {H}_{∞ }$$ mode reduction for two-dimensional discrete state-delayed systems, IEE Proceedings-Vision, Image and Signal Processing, 153, 769-784, (2006) [23] Wu, L; Shi, P; Gao, H; Wang, C, $$\mathscr {H}_{∞ }$$ filtering for 2-D Markovian jump systems, Automatica, 44, 1849-1858, (2008) · Zbl 1149.93346 [24] Wu, L; Zheng, W, Weighted $$\mathscr {H}_{∞ }$$ model reduction for linear switched systems with time-varying delay, Automatica, 45, 186-193, (2009) · Zbl 1154.93326 [25] Xie, L; Du, C; Soh, YC; Zhang, C, $${\mathscr {H}}_{∞ }$$ and robust control of 2-D systems in FM second model, Multidimensional Systems and Signal Processing, 13, 256-287, (2002) · Zbl 1028.93019 [26] Xu, H; Zou, Y; Xu, S; Lam, J; Wang, Q, $$\mathscr {H}_{∞ }$$ model reduction of 2-D singular Roesser models, Multidimensional Systems and Signal Processing, 16, 285-304, (2005) · Zbl 1219.93022 [27] Xu, S; Lam, J; Lin, Z; Galkowski, K, Positive real control for uncertain two-dimensional systems, IEEE Transactions on Circuits and Systems I: Regular Papers, 49, 1659-1666, (2002) · Zbl 1368.93193 [28] Xu, S; Lam, J; Lin, Z; Galkowski, K; Paszke, W; Sulikowski, B; etal., Positive real control of two-dimensional systems: Roesser models and linear repetitive processes, International Journal of Control, 76, 1047-1058, (2003) · Zbl 1049.93052 [29] Xu, S; Lam, J; Mao, X, Delay-dependent $$\mathscr {H}_{∞ }$$ control and filtering for uncertain Markovian jump systems with time-varying delays, IEEE Transactions on Circuits and Systems I: Regular Papers, 54, 2070-2077, (2007) · Zbl 1374.93134 [30] Xu, S; Lam, J; Zou, Y; Lin, Z; Paszke, W, Robust $${\mathscr {H}}_{∞ }$$ filtering for uncertain 2-D continuous systems, IEEE Transactions on Signal Processing, 53, 1731-1738, (2005) · Zbl 1370.93292 [31] Yan, W; Lam, J, An approximation approach to $$\mathscr {H}_{2}$$ optimal model reduction, IEEE Transactions on Automatic Control, 44, 1341-1358, (1999) · Zbl 0980.93011 [32] Zhang, L; Boukas, EK, Mode-dependent $$\mathscr {H}_{∞ }$$ filtering for discrete-time Markovian jump linear systems with partly unknown transition probabilities, Automatica, 45, 1462-1467, (2009) · Zbl 1166.93378 [33] Zhang, L; Huang, B; Lam, J, $$\mathscr {H}_{∞ }$$ model reduction of Markovian jump linear systems, Systems & Control Letters, 50, 103-118, (2003) · Zbl 1157.93519 [34] Zhao, Y; Zhang, L; Shen, S; Gao, H, Robust stability criterion for discrete-time uncertain Markovian jumping neural networks with defective statistics of modes transitions, IEEE Transactions on Neural Network, 22, 164-170, (2011) [35] Zhou, K, Frequency weighted $$\mathscr {L}_{∞ }$$ norm and optimal Hankel norm model reduction, IEEE Transactions on Automatic Control, 40, 1687-1699, (1995) · Zbl 0844.93022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.